Minimizing Price of Anarchy in Resource Allocation Games

178 Pages·2017·1.08 MB·English

University of Colorado, Boulder
CU Scholar
Electrical, Computer & Energy Engineering
Electrical, Computer & Energy Engineering
Graduate Theses & Dissertations
Spring 4-1-2014
Minimizing Price of Anarchy in Resource
Allocation Games
Yassmin Shalaby
University of Colorado Boulder, [email protected]
Follow this and additional works at:https://scholar.colorado.edu/ecen_gradetds
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Recommended Citation
Shalaby, Yassmin, "Minimizing Price of Anarchy in Resource Allocation Games" (2014).Electrical, Computer & Energy Engineering
Graduate Theses & Dissertations. 85.
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Minimizing Price of Anarchy in Resource Allocation Games
by
Yassmin Shalaby
B.S., Cairo University, 2006
M.S., University of Toronto, 2010
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Electrical, Computer, and Energy Engineering
2014
This thesis entitled:
Minimizing Price of Anarchy in Resource Allocation Games
written by Yassmin Shalaby
has been approved for the Department of Electrical, Computer, and Energy Engineering
Jason Marden
Eric Frew
Date
The final copy of this thesis has been examined by the signatories, and we find that both the
content and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
iii
Shalaby, Yassmin (M.S., Electrical, Computer, and Energy Engineering)
Minimizing Price of Anarchy in Resource Allocation Games
Thesis directed by Prof. Jason Marden
Resource allocation refers to problems where there is a set of resources to be allocated effi-
ciently among a group of agents. The distributed nature of resource allocation motivates modeling
it as a distributed control problem. One of the strong modeling frameworks for distributed control
problems is the game theoretic framework. Game theory provides mathematical models that aid
in studying the aggregate behavior of a group of decision makers. The main challenge in modeling
a distributed optimization problem as a game is the design of agents’ utility functions. A utility
function is designed as a distribution rule of some welfare; and the goal is to distribute the welfare
in a way that incentivizes players to land in a “good” equilibrium point. The ratio between the
performanceoftheworstpossibleequilibriumpointandtheoptimaloutcomeofagameiscalledthe
price of anarchy. A distribution rule that distributes the welfare exactly is called budget-balanced,
andonethatdistributesthewelfarewithexcessissaidtosatisfyarelaxedbudget-balancecondition.
On the other hand, if it causes a deficit we say that it violates the budget-balance condition.
In this thesis, we study the design of utility functions in resource allocation games that mini-
mize the price of anarchy. We compare two families of utility functions that guarantee equilibrium
existence, namely the Shapley value and the marginal contribution. The Shapley value is a budget-
balanced distribution rule, while the marginal contribution satisfies the relaxed budget-balance
condition given that the welfare being distributed is submodular. We derive price of anarchy
bounds for the marginal contribution utility in resource allocation games and compare them to
those for the Shapley value, derived in the literature. We also perform a small-scale study for a
wider range of utility functions. Lastly, we examine the connection between the price of anarchy
andthesatisfiabilityofthebudget-balanceconditionsoftheutilitydesigns. Weshowthatviolating
the budget-balance condition worsens the price of anarchy.
Dedication
To my beloved husband, Zyad.
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Acknowledgements
I would like to express deep gratitude towards my advisor, Jason Marden, for the guidance
and support he offered me throughout our research journey. I would like to greatly thank Raga
Gopalakrishnan for his insights on this research; and my thesis committee for reviewing my work.
I was blessed to have wonderful lab-mates, Holly, Yilan, Philip, Matt and many others who
havejoinedourlaboccasionally. Iamthankfulforthefruitfuldiscussionswehavehadoverresearch
and the enjoyable lunch and coffee breaks we have shared together.
WarmthankstomyfriendsAya,Samah,Heba,Hoda,Paria,Nancy,Khawla,Rehab,Khadija,
Kelly, Marie, Neveen, Reham and many others who have been by my side in the ups and downs.
I greatly acknowledge Sara Fall, at the graduate writing support center, for helping me in
editing my thesis. I would also like to acknowledge all the employees at the University of Colorado
atBoulder,fromgraduatestudentadvisors,tochefsandstaffattheC4CcafeteriaandBuff-busand
night-ride drivers, for making my life as a graduate student much easier. Special thanks to Adam,
my counselors Dorothy and Glenda and my thesis support group; Sara, Richard, and Charlie.
Words will never be enough to praise my family and friends back home who, though ge-
ographically far away, have always been very close to me, supporting and encouraging me. My
parents have always been my backbone and have given me the chance to fulfill my dreams and
excel in every way in life. My sister, brother and my in-laws have always been there for me.
I passionately thank my husband, Zyad, for all the sacrifices he did for me and for the time
he spent babysitting our son so that I can get work done. I am so grateful to have you in my life.
Thanks to my 3-years-old son for bringing much joy into my life and inspiring me in so many ways.
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All my success is due to Allah (God); in Him I trust and to Him I turn.
Contents
Chapter
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Background 11
2.1 Normal-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Types of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Existence of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Welfare Distribution Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Equal Share . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Marginal Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.3 Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 The weighted Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.1 Distribution Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.2 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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3 Related work 27
3.1 Valid Utility Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Smooth Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Potential Games Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Efficiency of Shapley Value in Resource Allocation Games . . . . . . . . . . . . . . . 33
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Marginal Contribution Utility Function in Resource Allocation Games 36
4.1 Game Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Price of Anarchy Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.1 Symmetric Action Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 Asymmetric Action Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.2 Analytical Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Worst Case Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.2 Shapley value worst case format . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.3 Marginal Contribution worst case format . . . . . . . . . . . . . . . . . . . . 69
4.4.4 Weighted Shapley value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Small-Scale Study on Utility functions that minimize the Price of Anarchy for Resource
Allocation Games 81
5.1 2-player Resource Allocation Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.1 Game Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.2 Game Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.3 Price of Anarchy Analysis for 2-player games . . . . . . . . . . . . . . . . . . 91
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5.1.4 Price of Stability Analysis for 2-player games . . . . . . . . . . . . . . . . . . 99
5.2 3-player Resource Allocation Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.1 Game Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.2 Price of Anarchy Analysis for 3-player games . . . . . . . . . . . . . . . . . . 107
5.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3 Resource Allocation Games with a finite number of players . . . . . . . . . . . . . . 119
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Effect of Violating the Budget-Balance Condition on the Robust Price of Anarchy 122
6.1 Reverse Carpooling Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.1.1 Game Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.1.2 Cost Distribution Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Non-Budget-Balanced Smooth Games . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.3 Smoothness of Reverse Carpooling Games . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7 Conclusions and Future Recommendations 142
7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Related Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Bibliography 147
Appendix
A Efficiency bounds 150
B Proof for some lemmas 152